Bureau International des Poids et Mesures. The International System of Units (SI)

Transcription

1 Bureau International des Poids et Mesures The International System of Units (SI) 9th edition 2019

2 2 Draft of the ninth SI Brochure, 5 February 2018 The BIPM and the Metre Convention The International Bureau of Weights and Measures (BIPM) was set up by the Metre Convention signed in Paris on 20 May 1875 by seventeen States during the final session of the diplomatic Conference of the Metre. This Convention was amended in The BIPM has its headquarters near Paris, in the grounds ( m 2 ) of the Pavillon de Breteuil (Parc de Saint-Cloud) placed at its disposal by the French Government; its upkeep is financed jointly by the Member States of the Metre Convention. The task of the BIPM is to ensure worldwide unification of measurements; its function is thus to: establish fundamental standards and scales for the measurement of the principal physical quantities and maintain the international prototypes; carry out comparisons of national and international standards; ensure the coordination of corresponding measurement techniques; carry out and coordinate measurements of the fundamental physical constants relevant to these activities. The BIPM operates under the exclusive supervision of the International Committee for Weights and Measures (CIPM) which itself comes under the authority of the General Conference on Weights and Measures (CGPM) and reports to it on the work accomplished by the BIPM. Delegates from all Member States of the Metre Convention attend the General Conference which, at present, meets every four years. The function of these meetings is to: discuss and initiate the arrangements required to ensure the propagation and improvement of the International System of Units (SI), which is the modern form of the metric system; confirm the results of new fundamental metrological determinations and various scientific resolutions of international scope; take all major decisions concerning the finance, organization and development of the BIPM. The CIPM has eighteen members each from a different State: at present, it meets every year. The officers of this committee present an annual report on the administrative and financial position of the BIPM to the Governments of the Member States of the Metre Convention. The principal task of the CIPM is to ensure worldwide uniformity in units of measurement. It does this by direct action or by submitting proposals to the CGPM. As of 20 May 2019, fiftyeight States were members of this Convention: Argentina, Australia, Austria, Belgium, Brazil, Bulgaria, Canada, Chile, China, Colombia, Croatia, Czech Republic, Denmark, Egypt, Finland, France, Germany, Greece, Hungary, India, Indonesia, Iran (Islamic Rep. of), Iraq, Ireland, Israel, Italy, Japan, Kazakhstan, Kenya, Korea (Republic of), Lithuania, Malaysia, Mexico, The Netherlands, New Zealand, Norway, Pakistan, Poland, Portugal, Romania, Russian Federation, Saudi Arabia, Serbia, Singapore, Slovakia, Slovenia, South Africa, Spain, Sweden, Switzerland, Thailand, Tunisia, Turkey, United Arab Emirates, United Kingdom, United States of America, Uruguay, and Venezuela (Bolivarian Rep. of). Forty-three States and Economies were Associates of the General Conference: Albania, Azerbaijan, Bangladesh, Belarus, Bolivia, Bosnia and Herzegovina, Botswana, CARICOM, Chinese Taipei, Costa Rica, Cuba, Ecuador, Estonia, Ehiopia, Georgia, Ghana, Hong Kong (China), Jamaica, Latvia, Luxembourg, Macedonia (fmr Yugoslav Rep. of), Malta, Mauritius, Moldova (Republic of), Mongolia, Montenegro, Namibia, Oman, Panama, Paraguay, Peru, Philippines, Qatar, Seychelles, Sri Lanka, Sudan, Syrian Arab Republic, Tanzania, united Republic of Ukraine, Viet Nam, Yemen, Zambia, and Zimbawe

3 Draft of the ninth SI Brochure, 5 February The activities of the BIPM, which in the beginning were limited to measurements of length and mass, and to metrological studies in relation to these quantities, have been extended to standards of measurement of electricity (1927), photometry and radiometry (1937), ionizing radiation (1960), time scales (1988) and to chemistry (2000). To this end the original laboratories, built in , were enlarged in 1929; new buildings were constructed in for the ionizing radiation laboratories, in 1984 for the laser work and in 1988 for a library and offices. In 2001 a new building for the workshop, offices and meeting rooms was opened. Some forty-five physicists and technicians work in the BIPM laboratories. They mainly conduct metrological research, international comparisons of realizations of units and calibrations of standards. An annual report, the Director s Report on the Activity and Management of the International Bureau of Weights and Measures, gives details of the work in progress. Following the extension of the work entrusted to the BIPM in 1927, the CIPM has set up bodies, known as Consultative Committees, whose function is to provide it with information on matters that it refers to them for study and advice. These Consultative Committees, which may form temporary or permanent working groups to study special topics, are responsible for coordinating the international work carried out in their respective fields and for proposing recommendations to the CIPM concerning units. The Consultative Committees have common regulations (BIPM Proc.-Verb. Com. Int. Poids et Mesures, 1963, 31, 97). They meet at irregular intervals. The president of each Consultative Committee is designated by the CIPM and is normally a member of the CIPM. The members of the Consultative Committees are metrology laboratories and specialized institutes, agreed by the CIPM, which send delegates of their choice. In addition, there are individual members appointed by the CIPM, and a representative of the BIPM (Criteria for membership of Consultative Committees, BIPM Proc.-Verb. Com. Int. Poids et Mesures, 1996, 64, 124). At present, there are ten such committees: 1. The Consultative Committee for Electricity and Magnetism (CCEM), new name given in 1997 to the Consultative Committee for Electricity (CCE) set up in 1927; 2. The Consultative Committee for Photometry and Radiometry (CCPR), new name given in 1971 to the Consultative Committee for Photometry (CCP) set up in 1933 (between 1930 and 1933 the CCE dealt with matters concerning photometry); 3. The Consultative Committee for Thermometry (CCT), set up in 1937; 4. The Consultative Committee for Length (CCL), new name given in 1997 to the Consultative Committee for the Definition of the Metre (CCDM), set up in 1952; 5. The Consultative Committee for Time and Frequency (CCTF), new name given in 1997 to the Consultative Committee for the Definition of the Second (CCDS) set up in 1956; 6. The Consultative Committee for Ionizing Radiation (CCRI), new name given in 1997 to the Consultative Committee for Standards of Ionizing Radiation (CCEMRI) set up in 1958 (in 1969 this committee established four sections: Section I (X- and γ-rays, electrons), Section II (Measurement of radionuclides), Section III (Neutron measurements), Section IV (α-energy standards); in 1975 this last section was dissolved and Section II was made responsible for its field of activity); 7. The Consultative Committee for Units (CCU), set up in 1964 (this committee replaced the Commission for the System of Units set up by the CIPM in 1954);

4 4 Draft of the ninth SI Brochure, 5 February The Consultative Committee for Mass and Related Quantities (CCM), set up in 1980; 9. The Consultative Committee for Amount of Substance: Metrology in Chemistry and Biology (CCQM), set up in 1993; 10. The Consultative Committee for Acoustics, Ultrasound and Vibration (CCAUV), set up un The proceedings of the General Conference and the CIPM are published by the BIPM in the following series: Report of the meeting of the General Conference on Weights and Measures; Report of the meeting of the International Committee for Weights and Measures. The CIPM decided in 2003 that the reports of meetings of the Consultative Committees should no longer be printed, but would be placed on the BIPM website, in their original language. The BIPM also publishes monographs on special metrological subjects and, under the title The International System of Units (SI), this brochure, periodically updated, in which are collected all the decisions and recommendations concerning units. The collection of the Travaux et Mémoires du Bureau International des Poids et Mesures (22 volumes published between 1881 and 1966) and the Recueil de Travaux du Bureau International des Poids et Mesures (11 volumes published between 1966 and 1988) ceased by a decision of the CIPM. The scientific work of the BIPM is published in the open scientific literature and an annual list of publications appears in the Director s Report on the Activity and Management of the International Bureau of Weights and Measures. Since 1965 Metrologia, an international journal published under the auspices of the CIPM, has printed articles dealing with scientific metrology, improvements in methods of measurement, work on standards and units, as well as reports concerning the activities, decisions and recommendations of the various bodies created under the Metre Convention.

5 Draft of the ninth SI Brochure, 5 February The International System of Units Contents The BIPM and the Metre Convention 2 Preface to the 9th edition 7 1 Introduction The SI and the defining constants Motivation for the use of defining constants to define the SI Implementation of the SI 9 2 The International System of Units Defining the unit of a quantity Definition of the SI The nature of the seven defining constants Definitions of the SI units Base units Practical realization of SI units Dimensions of quantities Derived units Units for quantities that describe biological and physiological effects SI units in the framework of the general theory of relativity 24 3 Decimal multiples and sub-multiples of SI units 26 4 Non-SI units that are accepted for use with the SI 28 5 Writing unit symbols and names, and expressing the values of quantities The use of unit symbols and names Unit symbols Unit names Rules and style conventions for expressing values of quantities Value and numerical value of a quantity, and the use of quantity calculus Quantity symbols and unit symbols Formatting the value of a quantity Formatting numbers, and the decimal marker Expressing the measurement uncertainty in the value of a quantity Multiplying or dividing quantity symbols, the values of quantities, or numbers Stating quantity values being pure numbers Plane angles, solid angles and phase angles 34

6 6 Draft of the ninth SI Brochure, 5 February 2018 Appendix 1. Decisions of the CGPM and the CIPM 35 Appendix 2. Practical realization of the definitions of some Important units 35 Appendix 3. Units for photochemical and photobiological quantities 35 Appendix 4. Historical notes on the development of the International System of Units and its base units 35 Part 1. The historical development of the realization of SI units 35 Part 2. The historical development of the International System 37 Part 3. Historical perspective on the base units 40 List of acronyms Index XXX XXX

7 Draft of the ninth SI Brochure, 5 February Preface to the 9 th edition (to be completed)

8 8 Draft of the ninth SI Brochure, 5 February Introduction 1.1 The SI and the defining constants This brochure presents information on the definition and use of the International System of Units, universally known as the SI (from the French Système international d unités), for which the General Conference on Weights and Measures (CGPM) has responsibility. In 1960 the 11th CGPM formally defined and established the SI and has subsequently revised it from time to time in response to the requirements of users and advances in science and technology. The most recent and perhaps the most significant revision of the SI since its establishment was made by the 26th CGPM (2018) and is documented in this 9th edition of the SI Brochure. The Metre Convention and its organs, the CGPM, the Comité International des Poids et Mesures (CIPM), the Bureau International des Poids et Mesures (BIPM), and the Consultative Committees are described in the preface. The SI is a consistent system of units for use in all aspects of life, including international trade, manufacturing, security, health and safety, protection of the environment, and in the basic science that underpins all of these. The system of quantities underlying the SI and the equations relating them are based on the present description of nature and are familiar to all scientists, technologists and engineers. The definition of the SI units is established in terms of a set of seven defining constants. The complete system of units can be derived from the fixed values of these defining constants, expressed in the units of the SI. These seven defining constants are the most fundamental feature of the definition of the entire system of units. These particular constants were chosen after having been identified as being the best choice, taking into account the previous definition of the SI, which is based on seven base units, and progress in science. A variety of experimental methods described by the Consultative Committees of the International Committee for Weights and Measures (CIPM) may be used to realize the definitions. Descriptions of these realizations are also referred to as mises en pratique. Realizations may be revised whenever new experiments are developed; for this reason advice on realizing the definitions is not included in this brochure but is available from the BIPM. 1.2 Motivation for the use of defining constants to define the SI Historically, SI units have been presented in terms of a set of most recently seven base units. All other units, described as derived units, are constructed as products of powers of the base units. Different types of definitions for the base units have been used: artefacts such as the international prototype (IPK) for the unit kilogram; a specific physical state such as the triple point of water for the unit kelvin; idealized experimental prescriptions as in the case of the ampere and the candela; or constants of nature such as the speed of light for the definition of the unit metre.

9 Draft of the ninth SI Brochure, 5 February To be of any practical use, these units not only have to be defined, but they also have to be realized physically for dissemination. In the case of an artefact, the definition and the realization are equivalent a path that was pursued by advanced ancient civilizations. Although this is simple and clear, artefacts involve the risk of loss, damage or change. The other types of unit definitions are increasingly abstract or idealized. Here, the realizations are separated conceptually from the definitions so that the units can, as a matter of principle, be realized independently at any place and at any time. In addition, new and superior realizations may be introduced as science and technologies develop, without the need to redefine the unit. These advantages most obviously seen with the history of the definition of the metre from artefacts through an atomic reference transition to the fixed numerical value of the speed of light led to the decision to define all units by using defining constants. The choices of the base units were never unique, but grew historically and became familiar to users of the SI. This description in terms of base and derived units is maintained in the present definition of the SI, but has been reformulated as a consequence of adoption of the defining constants. 1.3 Implementation of the SI The definitions of the SI units, as decided by the CGPM, represent the highest reference level for measurement traceability to the SI. Metrology institutes around the world establish the practical realizations of the definitions in order to allow for traceability of measurements to the SI. The Consultative Committees provide the framework for establishing the equivalence of the realizations in order to harmonize traceability world-wide. Standardization bodies may specify further details for quantities and units and rules for their application, where these are needed by interested parties. Whenever SI units are involved, these standards must refer to the definitions by the CGPM. Many such specifications are listed for example in the International Organization for Standardization International Electrotechnical Commission (ISO/ IEC series of international standards). Individual countries have established rules concerning the use of units by national legislation, either for general use or for specific areas such as commerce, health, public safety and education. In almost all countries, this legislation is based on the SI. The International Organization of Legal Metrology (OIML) is charged with the international harmonization of the technical specifications of this legislation.

10 10 Draft of the ninth SI Brochure, 5 February The International System of Units 2.1 Defining the unit of a quantity The value Q of a quantity is expressed by the product of a number {Q} and a unit [Q]: Q = {Q}[Q]. The unit is simply a particular example of the value of a quantity, defined by convention, which is used as a reference and the number is the ratio of the value of the quantity to the unit. For a particular quantity different units may be used. For example, the value of the speed v of a particle may be expressed as v = 25 m/s or v = 90 km/h, where metre per second and kilometre per hour are alternative units for the same value of the quantity speed. Before stating the result of a measurement, it is essential that the quantity being presented is adequately described. This may be simple, as in the case of the length of a particular steel rod, but can become more complex when higher accuracy is required and where additional parameters, such as temperature, need to be specified. When a measurement result of a quantity is reported, the estimated value of the measurand (the quantity to be measured), and the uncertainty associated with that value, are necessary. Both are expressed in the same unit. 2.2 Definition of the SI As for any quantity, the value of a fundamental constant can be expressed as the product of a number and a unit as Q = {Q} [Q]. The definitions below specify the exact numerical value of each constant when its value is expressed in the corresponding SI unit. By fixing the exact numerical value the unit becomes defined, since the product of the numerical value {Q} and the unit [Q] has to equal the value Q of the constant, which is postulated to be invariant. The seven constants are chosen in such a way that any unit of the SI can be written either through a defining constant itself or through products or ratios of defining constants. The International System of Units, the SI, is the system of units in which the unperturbed ground state hyperfine transition frequency of the caesium 133 atom ν Cs is Hz, the speed of light in vacuum c is m/s, the Planck constant h is J s, the elementary charge e is C, the Boltzmann constant k is J/K, the Avogadro constant N A is mol 1, the luminous efficacy of monochromatic radiation of frequency hertz K cd is 683 lm/w.

11 Draft of the ninth SI Brochure, 5 February where the hertz, joule, coulomb, lumen, and watt, with unit symbols Hz, J, C, lm, and W, respectively, are related to the units second, metre, kilogram, ampere, kelvin, mole, and candela, with unit symbols s, m, kg, A, K, mol, and cd, respectively, according to Hz = s 1, J = m 2 kg s 2, C = A s, lm = cd m 2 m 2 = cd sr, and W = m 2 kg s 3 The numerical values of the seven defining constants have no uncertainty. Table 1. The seven defining constants of the SI and the seven corresponding units they define Defining constant Symbol Numerical value Unit hyperfine transition frequency of Cs ν Cs Hz speed of light in vacuum c m s 1 Planck constant h J s elementary charge e C Boltzmann constant k J K 1 Avogadro constant N A mol 1 luminous efficacy K cd 683 lm W 1 Preserving continuity, as far as possible, has always been an essential feature of any changes to the International System of Units. The numerical values of the defining constants have been chosen to be consistent with the earlier definitions in so far as advances in science and knowledge allow The nature of the seven defining constants The nature of the defining constants ranges from fundamental constants of nature to technical constants. The use of a constant to define a unit disconnects definition from realization. This offers the possibility that completely different or new and superior practical realizations can be developed, as technologies evolve, without the need to change the definition. A technical constant such as luminous efficacy (K cd ) refers to a special application. It can, in principle, be chosen freely, such as to include conventional physiological or other weighting factors. In contrast, the use of a fundamental constant of nature, in general, does not allow this choice because it is related to other constants through the equations of physics. The set of seven defining constants has been chosen to provide a fundamental, stable and universal reference that simultaneously allows for practical realizations with the smallest uncertainties. The technical conventions and specifications also take historical developments into account. Both the Planck constant h and the speed of light in vacuum c are properly described as fundamental. They determine quantum effects and space-time properties, respectively, and affect all particles and fields equally on all scales and in all environments.

12 12 Draft of the ninth SI Brochure, 5 February 2018 The elementary charge e corresponds to a coupling strength of the electromagnetic force via the fine-structure constant α = e 2 /(2cε 0 h) where ε 0 is the vacuum electric permittivity or electric constant. Some theories predict a variation of α over time. The experimental limits of the maximum possible variation in α are so low, however, that any effect on foreseeable practical measurements can be excluded. The Boltzmann constant k corresponds to a conversion factor between the quantities temperature (with unit kelvin) and energy (with unit joule), whereby the numerical value is obtained from historical specifications of the temperature scale. The temperature of a system scales with the thermal energy, but not necessarily with the internal energy of a system. In statistical physics the Boltzmann constant connects the entropy S with the number Ω of quantum-mechanically accessible states, S = k ln Ω. The caesium frequency ν Cs, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, has the character of an atomic parameter, which may be affected by the environment, such as electromagnetic fields. However, the underlying transition is well understood, stable and a good choice as a reference transition under practical considerations. The choice of an atomic parameter like ν Cs does not disconnect definition and realization in the same way that h, c, e, or k do, but specifies the reference. The Avogadro constant N A corresponds to a conversion factor between the quantity amount of substance (with unit mole) and the quantity for counting entities (with unit one, symbol 1). Thus it has the character of a constant of proportionality similar to the Boltzmann constant k. The luminous efficacy K cd is a technical constant that gives an exact numerical relationship between the purely physical characteristics of the radiant power stimulating the human eye (W) and its photobiological response defined by the luminous flux due to the spectral responsivity of a standard observer (lm) at a frequency of hertz. 2.3 Definitions of the SI units Prior to the definitions adopted in 2018, the SI was defined through seven base units from which the derived units were constructed as products of powers of the base units. Defining the SI by fixing the numerical values of seven defining constants has the effect that this distinction is, in principle, not needed, since all units, base as well as derived units, may be constructed directly from the defining constants. Nevertheless, the concept of base and derived units is maintained, not only because it is useful and historically well established, but also because it is necessary to maintain consistency with the International System of Quantities (ISQ) defined by the ISO/IEC series of Standards, which specify base and derived quantities to which the SI base and derived units necessarily correspond.

13 Draft of the ninth SI Brochure, 5 February Base units The base units of the SI are listed in Table 2. Table 2. SI base units Base quantity Base unit Name Typical symbol Name Symbol time t second s length l, x, r, etc. metre m mass m kilogram kg electric current I, i ampere A thermodynamic temperature T kelvin K amount of substance n mole mol luminous intensity I v candela cd The symbols for quantities are generally single letters of the Latin or Greek alphabets, printed in an italic font, and are recommendations. The symbols for units are printed in an upright (roman) font and are mandatory, see chapter 5. Starting from the definition of the SI in terms of fixed numerical values of the defining constants, definitions of each of the seven base units are deduced by using, as appropriate, one or more of these defining constants to give the following set of definitions: The second The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency ν Cs, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be when expressed in the unit Hz, which is equal to s 1. This definition implies the exact relation ν Cs = Hz. Inverting this relation gives an expression for the unit second in terms of the value of the defining constant ν Cs : ν Cs 1Hz = or s =. ν Cs The effect of this definition is that the second is equal to the duration of periods of the radiation corresponding to the transition between the two hyperfine levels of the unperturbed ground state of the 133 Cs atom. The reference to an unperturbed atom is intended to make it clear that the definition of the SI second is based on an isolated caesium atom that is unperturbed by any external field, such as ambient black-body radiation. The second, so defined, is the unit of proper time in the sense of the general theory of relativity. To allow the provision of a coordinated time scale, the signals of different primary clocks in different locations are combined, which have to be corrected for relativistic caesium frequency shifts (see section 2.3.6). The CIPM has adopted various secondary representations of the second, based on a selected number of spectral lines of atoms, ions or molecules. The unperturbed frequencies of these lines can be determined with a relative uncertainty not lower than that of the realization of the second based on the 133 Cs hyperfine transition frequency, but some can be reproduced with superior stability.

14 14 Draft of the ninth SI Brochure, 5 February 2018 The metre The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be when expressed in the unit m s 1, where the second is defined in terms of the caesium frequency ν Cs. This definition implies the exact relation c = m s 1. Inverting this relation gives an exact expression for the metre in terms of the defining constants c and ν Cs : 1m = c s = c c ν ν Cs Cs. The effect of this definition is that one metre is the length of the path travelled by light in vacuum during a time interval with duration of 1/ of a second. The kilogram The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be when expressed in the unit J s, which is equal to kg m 2 s 1, where the metre and the second are defined in terms of c and ν Cs. This definition implies the exact relation h = kg m 2 s 1. Inverting this relation gives an exact expression for the kilogram in terms of the three defining constants h, ν Cs and c: h 2 1kg m s = which is equal to 1kg = 2 ( ) h ν Cs 40 h ( )( ) c c ν Cs The effect of this definition is to define the unit kg m 2 s 1 (the unit of both the physical quantities action and angular momentum). Together with the definitions of the second and the metre this leads to a definition of the unit of mass expressed in terms of the value of the Planck constant h. The previous definition of the kilogram fixed the value of the mass of the international prototype of the kilogram, m(k), to be equal to one kilogram exactly and the value of the Planck constant h had to be determined by experiment. The present definition fixes the numerical value of h exactly and the mass of the prototype has now to be determined by experiment. The number chosen for the numerical value of the Planck constant in this definition is such that at the time of its adoption, the kilogram was equal to the mass of the international prototype, m(k) = 1 kg, with a relative standard uncertainty of , which was the standard uncertainty of the combined best estimates of the value of the Planck constant at that time. Note that with the present definition, primary realizations can be established, in principle, at any point in the mass scale.

15 Draft of the ninth SI Brochure, 5 February The ampere The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be when expressed in the unit C, which is equal to A s, where the second is defined in terms of ν Cs. This definition implies the exact relation e = A s. Inverting this relation gives an exact expression for the unit ampere in terms of the defining constants e and ν Cs : e 1 1A s = which is equal to 1 8 1A ν e ν e Cs = 19 Cs 10 ( )( ) The effect of this definition is that one ampere is the electric current corresponding to the flow of 1/( ) elementary charges per second. The previous definition of the ampere was based on the force between two current carrying conductors and had the effect of fixing the value of the vacuum magnetic permeability μ 0 (also known as the magnetic constant) to be exactly 4π 10 7 H m 1 = 4π 10 7 N A 2, where H and N denote the coherent derived units henry and newton, respectively. The new definition of the ampere fixes the value of e instead of μ 0. As a result, μ 0 must be determined experimentally. It also follows that since the vacuum electric permittivity ε 0 (also known as the electric constant), the characteristic impedance of vacuum Z 0, and the admittance of vacuum Y 0 are equal to 1/μ 0 c 2, μ 0 c, and 1/μ 0 c, respectively, the values of ε 0, Z 0, and Y 0 must now also be determined experimentally, and are affected by the same relative standard uncertainty as μ 0 since c is exactly known. The product ε 0 μ 0 = 1/c 2 and quotient Z 0 /μ 0 = c remain exact. At the time of adopting the present definition of the ampere, µ 0 was equal to 4π 10 7 H/m with a relative standard uncertainty of The kelvin The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by taking the fixed numerical value of the Boltzmann constant k to be when expressed in the unit J K 1, which is equal to kg m 2 s 2 K 1, where the kilogram, metre and second are defined in terms of h, c and ν Cs. This definition implies the exact relation k = kg m 2 s 2 K 1. Inverting this relation gives an exact expression for the kelvin in terms of the defining constants k, h and ν Cs : K = k kg m 2 s 2

16 16 Draft of the ninth SI Brochure, 5 February 2018 which is equal to 1K = 23 ν Csh ν ( )( ) k k The effect of this definition is that one kelvin is equal to the change of thermodynamic temperature that results in a change of thermal energy kt by J. The previous definition of the kelvin set the temperature of the triple point of water, T TPW, to be exactly K. Due to the fact that the present definition of the kelvin fixes the numerical value of k instead of T TPW, the latter must now be determined experimentally. At the time of adopting the present definition T TPW was equal to K with a relative standard uncertainty of based on measurements of k made prior to the redefinition. As a result of the way temperature scales used to be defined, it remains common practice to express a thermodynamic temperature, symbol T, in terms of its difference from the reference temperature T 0 = K, close to the ice point. This difference is called the Celsius temperature, symbol t, which is defined by the quantity equation t = T T 0. The unit of Celsius temperature is the degree Celsius, symbol C, which is by definition equal in magnitude to the unit kelvin. A difference or interval of temperature may be expressed in kelvin or in degrees Celsius, the numerical value of the temperature difference being the same in either case. However, the numerical value of a Celsius temperature expressed in degrees Celsius is related to the numerical value of the thermodynamic temperature expressed in kelvin by the relation t/ C = T/K (see for an explanation of the notation used here). The kelvin and the degree Celsius are also units of the International Temperature Scale of 1990 (ITS-90) adopted by the CIPM in 1989 in Recommendation 5 (CI-1989, PV, 57, 115). Note that the ITS-90 defines two quantities T 90 and t 90 which are close approximations to the corresponding thermodynamic temperatures T and t. Note that with the present definition, primary realizations of the kelvin can, in principle, be established at any point of the temperature scale. Cs h

17 Draft of the ninth SI Brochure, 5 February The mole The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly elementary entities. This number is the fixed numerical value of the Avogadro constant, N A, when expressed in the unit mol -1 and is called the Avogadro number. The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles. This definition implies the exact relation N A = mol 1. Inverting this relation gives an exact expression for the mole in terms of the defining constant N A : mol = N A The effect of this definition is that the mole is the amount of substance of a system that contains specified elementary entities. The previous definition of the mole fixed the value of the molar mass of carbon 12, M( 12 C), to be exactly kg/mol. According to the present definition M( 12 C) is no longer known exactly and must be determined experimentally. The value chosen for N A is such that at the time of adopting the present definition of the mole, M( 12 C) was equal to kg/mol with a relative standard uncertainty of The molar mass of any atom or molecule X may still be obtained from its relative atomic mass from the equation M(X) = A r (X) [M( 12 C)/12] = A r (X) M u and the molar mass of any atom or molecule X is also related to the mass of the elementary entity m(x) by the relation M(X) = N A m(x) = N A A r (X) m u. In these equations M u is the molar mass constant, equal to M( 12 C)/12 and m u is the unified atomic mass constant, equal to m( 12 C)/12. They are related to the Avogadro constant through the relation M u = N A m u. In the name amount of substance, the word substance will typically be replaced by words to specify the substance concerned in any particular application, for example amount of hydrogen chloride, HCl, or amount of benzene, C 6 H 6. It is important to give a precise definition of the entity involved (as emphasized in the definition of the mole); this should preferably be done by specifying the molecular chemical formula of the material involved. Although the word amount has a more general dictionary definition, the abbreviation of the full name amount of substance to amount may be used for brevity. This also applies to derived quantities such as amount-of-substance concentration, which may simply be called amount concentration. In the field of clinical chemistry, the name amount-of-substance concentration is generally abbreviated to substance concentration.

18 18 Draft of the ninth SI Brochure, 5 February 2018 The candela The candela, symbol cd, is the SI unit of luminous intensity in a given direction. It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency Hz, Kcd, to be 683 when expressed in the unit lm W 1, which is equal to cd sr W 1, or cd sr kg 1 m 2 s 3, where the kilogram, metre and second are defined in terms of h, c and ν Cs. This definition implies the exact relation K cd = 683 cd sr kg 1 m 2 s 3 for monochromatic radiation of frequency ν = Hz. Inverting this relation gives an exact expression for the candela in terms of the defining constants K cd, h and ν Cs : which is equal to cd 1cd = K kg m cd = ( ν 2 Cs ) h K cd s 3 sr 1 34 ( )( ) ( ν h K ) 2 Cs cd The effect of this definition is that one candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency Hz and has a radiant intensity in that direction of (1/683) W/sr. The definition of the steradian is given below Table Practical realization of SI units The highest-level experimental methods used for the realization of units using the equations of physics are known as primary methods. The essential characteristic of a primary method is that it allows a quantity to be measured in a particular unit by using only measurements of quantities that do not involve that unit. In the present formulation of the SI, the basis of the definitions is different from that used previously, so that new methods may be used for the practical realization of SI units. Instead of each definition specifying a particular condition or physical state, which sets a fundamental limit to the accuracy of realization, a user is now free to choose any convenient equation of physics that links the defining constants to the quantity intended to be measured. This is a much more general way of defining the basic units of measurement. It is not limited by today s science or technology; future developments may lead to different ways of realizing units to a higher accuracy. When defined this way, there is, in principle, no limit to the accuracy with which a unit might be realized. The exception remains the definition of the second, in which the original microwave transition of caesium must remain, for the time being, the basis of the definition. For a more comprehensive explanation of the realization of SI units see Appendix 2.

19 Draft of the ninth SI Brochure, 5 February Dimensions of quantities Physical quantities can be organized in a system of dimensions, where the system used is decided by convention. Each of the seven base quantities used in the SI is regarded as having its own dimension. The symbols used for the base quantities and the symbols used to denote their dimension are shown in Table 3. Table 3. Base quantities and dimensions used in the SI Base quantity Typical symbol for quantity Symbol for dimension time t T length l, x, r, etc. L mass m M electric current I, i I thermodynamic temperature T Θ amount of substance n N luminous intensity I v J All other quantities, with the exception of counts, are derived quantities, which may be written in terms of base quantities according to the equations of physics. The dimensions of the derived quantities are written as products of powers of the dimensions of the base quantities using the equations that relate the derived quantities to the base quantities. In general the dimension of any quantity Q is written in the form of a dimensional product, dim Q = T α L β M γ I δ Θ ε N ζ J η where the exponents α, β, γ, δ, ε, ζ and η, which are generally small integers, which can be positive, negative, or zero, are called the dimensional exponents. There are quantities Q for which the defining equation is such that all of the dimensional exponents in the equation for the dimension of Q are zero. This is true in particular for any quantity that is defined as the ratio of two quantities of the same kind. For example, the refractive index is the ratio of two speeds and the relative permittivity is the ratio of the permittivity of a dielectric medium to that of free space. Such quantities are simply numbers. The associated unit is the unit one, symbol 1, although this is rarely explicitly written (see 5.4.7). There are also some quantities that cannot be described in terms of the seven base quantities of the SI, but have the nature of a count. Examples are a number of molecules, a number of cellular or biomolecular entities (for example copies of a particular nucleic acid sequence), or degeneracy in quantum mechanics. Counting quantities are also quantities with the associated unit one. The unit one is the neutral element of any system of units necessarily and present automatically. There is no requirement to introduce it formally by decision. Therefore, a formal traceability to the SI can be established through appropriate, validated measurement procedures.

20 20 Draft of the ninth SI Brochure, 5 February 2018 Plane and solid angles, when expressed in radians and steradians respectively, are in effect also treated within the SI as quantities with the unit one (see section 5.4.8). The symbols rad and sr are written explicitly where appropriate, in order to emphasize that, for radians or steradians, the quantity being considered is, or involves the plane angle or solid angle respectively. For steradians it emphasizes the distinction between fluxes and intensities in radiometry and photometry for example. However, it is a long-established practice in mathematics and across all areas of science to make use of rad = 1 and sr = 1. For historical reasons the radian and steradian are treated as derived units, as described in section It is especially important to have a clear description of any quantity with unit one (see section 5.4.7) that is expressed as a ratio of quantities of the same kind (for example length ratios or amount fractions) or as a count (for example number of photons or decays) Derived units Derived units are defined as products of powers of the base units. When the numerical factor of this product is one, the derived units are called coherent derived units. The base and coherent derived units of the SI form a coherent set, designated the set of coherent SI units. The word coherent here means that equations between the numerical values of quantities take exactly the same form as the equations between the quantities themselves. Some of the coherent derived units in the SI are given special names. Table 4 lists 22 SI units with special names. Together with the seven base units (Table 2) they form the core of the set of SI units. All other SI units are combinations of some of these 29 units. It is important to note that any of the seven base units and 22 SI units with special names can be constructed directly from the seven defining constants. In fact, the units of the seven defining constants include both base and derived units. The CGPM has adopted a series of prefixes for use in forming the decimal multiples and sub-multiples of the coherent SI units (see chapter 3). They are convenient for expressing the values of quantities that are much larger than or much smaller than the coherent unit. However, when prefixes are used with SI units, the resulting units are no longer coherent, because the prefix introduces a numerical factor other than one. Prefixes may be used with any of the 29 SI units with special names with the exception of the base unit kilogram, which is further explained in chapter 3. Table 4. The 22 SI units with special names and symbols Special name Unit expressed in Unit expressed in Derived quantity of unit terms of base units (a) terms of other SI units plane angle radian (b) rad = m/m solid angle steradian (c) sr = m 2 /m 2 frequency hertz (d) Hz = s 1 force newton N = kg m s 2 pressure, stress pascal Pa = kg m 1 s 2 energy, work, joule J = kg m 2 s 2 N m amount of heat power, radiant flux watt W = kg m 2 s 3 J/s electric charge coulomb C = A s

21 Draft of the ninth SI Brochure, 5 February electric potential difference (e) volt V = kg m 2 s 3 A 1 W/A capacitance farad F = kg 1 m 2 s 4 A 2 C/V electric resistance ohm Ω = kg m 2 s 3 A 2 V/A electric conductance siemens S = kg 1 m 2 s 3 A 2 A/V magnetic flux weber Wb = kg m 2 s 2 A 1 V s magnetic flux density tesla T = kg s 2 A 1 Wb/m 2 inductance henry H = kg m 2 s 2 A 2 Wb/A Celsius temperature degree Celsius (f) o C = K luminous flux lumen lm = cd sr cd sr illuminance lux lx = cd sr m 2 lm/m 2 activity referred to becquerel Bq = s 1 (d, g) a radionuclide absorbed dose, kerma gray Gy = m 2 s 2 J/kg dose equivalent sievert (h) Sv = m 2 s 2 J/kg catalytic activity katal kat = mol s 1 (a) The order of symbols for base units in this Table is different from that in the 8th edition following a decision by the CCU at its 21st meeting (2013) to return to the original order in Resolution 12 of the 11th CGPM (1960) in which newton was written kg m s 2, the joule as kg m 2 s 2 and J s as kg m 2 s 1. The intention was to reflect the underlying physics of the corresponding quantity equations although for some more complex derived units this may not be possible. (b) The radian is the coherent unit for plane angle. One radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius. It is also the unit for phase angle. For periodic phenomena, the phase angle increases by 2π rad in one period. The radian was formerly an SI supplementary unit, but this category was abolished in (c) The steradian is the coherent unit for solid angle. One steradian is the solid angle subtended at the centre of a sphere by an area of the surface that is equal to the squared radius. Like the radian, the steradian was formerly an SI supplementary unit. (d) The hertz shall only be used for periodic phenomena and the becquerel shall only be used for stochastic processes in activity referred to a radionuclide. (e) Electric potential difference is also called voltage in many countries, as well as electric tension or simply tension in some countries. (f) The degree Celsius is used to express Celsius temperatures. The numerical value of a temperature difference or temperature interval is the same when expressed in either degrees Celsius or in kelvin. (g) Activity referred to a radionuclide is sometimes incorrectly called radioactivity. (h) See CIPM Recommendation 2 on the use of the sievert (PV, 2002, 70, 205). The seven base units and 22 units with special names and symbols may be used in combination to express the units of other derived quantities. Since the number of quantities is without limit, it is not possible to provide a complete list of derived quantities and derived units. Table 5 lists some examples of derived quantities and the corresponding coherent derived units expressed in terms of base units. In addition, Table 6 lists examples of coherent derived units whose names and symbols also include derived units. The complete set of SI units includes both the coherent set and the multiples and sub-multiples formed by using the SI prefixes.

22 22 Draft of the ninth SI Brochure, 5 February 2018 Table 5. Examples of coherent derived units in the SI expressed in terms of base units Derived quantity Typical symbol Derived unit expressed of quantity in terms of base units area A m 2 volume V m 3 speed, velocity v m s 1 acceleration a m s 2 wavenumber σ m 1 density, mass density ρ kg m 3 surface density ρ A kg m 2 specific volume v m 3 kg 1 current density j A m 2 magnetic field strength H A m 1 amount of substance concentration c mol m 3 mass concentration ρ, γ kg m 3 luminance L v cd m 2 Table 6. Examples of SI coherent derived units whose names and symbols include SI coherent derived units with special names and symbols Derived unit expressed Derived quantity Name of coherent derived unit Symbol in terms of base units dynamic viscosity pascal second Pa s kg m 1 s 1 moment of force newton metre N m kg m 2 s 2 surface tension newton per metre N m 1 kg s 2 angular velocity, radian per second rad s 1 s 1 angular frequency angular acceleration radian per second squared rad/s 2 s 2 heat flux density, watt per square metre W m 2 kg s 3 irradiance heat capacity, entropy joule per kelvin J K 1 kg m 2 s 2 K 1 specific heat capacity, joule per kilogram kelvin J K 1 kg 1 m 2 s 2 K 1 specific entropy specific energy joule per kilogram J kg 1 m 2 s 2 thermal conductivity watt per metre kelvin W m 1 K 1 kg m s 3 K 1 energy density joule per cubic metre J m 3 kg m 1 s 2 electric field strength volt per metre V m 1 kg m s 3 A 1 electric charge density coulomb per cubic metre C m 3 A s m 3 surface charge density coulomb per square metre C m 2 A s m 2 electric flux density, coulomb per square metre C m 2 A s m 2 electric displacement permittivity farad per metre F m 1 kg 1 m 3 s 4 A 2 permeability henry per metre H m 1 kg m s 2 A 2